Nonlinear Elliptic Equations on Expanding Symmetric Domains

AbstractIn this article we study the problem --EQUATION OMITTED-- in the case Ωa is an expanding domain. In particular, for n⩾2 when Ωa={x∈Rn:a<|x|<a+1} is an expanding annulus as a→∞, we prove the existence of many rotationally non-equivalent solutions obtained as local minimizers of the corresponding energy functional. Moreover, we study the exact symmetry and the shape of these solutions, and under certain conditions we prove the existence of solutions with prescribed symmetry

Enhanced Leak Detection

A key requirement for Veeder-Root’s Enhanced Leak Detection System is that it be able to test in situ for the presence of leaks at gasoline dispensing facilities. Aside from the obvious issues of safety and lost product, this functionality is obligatory for compliance with environmental standards mandated by federal and state oversight bodies, such as the California State Water Resources Control Board (SWRCB). The SWRCB demands a testing procedure that includes conditions as close to operational as possible, while still using environmentally safe gases as a test fluid. Although the test parameters (e.g., pressure) are allowed to deviate from operating conditions in order to facilitate the test procedure, a prescribed rescaling of the test thresholds must then be applied to account for the deviation. Whether the test is run at operation conditions or in a slightly different parameter regime, the fact that the testing must be done on the product and return lines after installation at a service station presents significant challenges in devising an effective test strategy

Positive Solutions Obtained as Local Minima via Symmetries, for Nonlinear Elliptic Equations

In this dissertation, we establish existence and multiplicity of positive solutions for semilinear elliptic equations with subcritical and critical nonlinearities. We treat problems invariant under subgroups of the orthogonal group. Roughly speaking, we prove that if enough mass is concentrated around special orbits, then among the functions with prescribed symmetry, there is a solution for the original problem. Our results can be regarded as a further development of the work of Z.-Q. Wang, where existence of local minima in the space of symmetric functions was studied for the Schrödinger equation. We illustrate the general theory with three examples, all of which produce new results. Our method allows the construction of solutions with prescribed symmetry, and it represents a step further in the classification of positive solutions for certain nonlinear elliptic problems

Sharp weighted-norm inequalities for functions with compact support in RN∖{0}

AbstractIn this paper we study a class of Caffarelli–Kohn–Nirenberg inequalities without restricting the pertinent parameters. In particular, we determine the values of the corresponding optimal constants and the functions that achieve them, i.e., minimizers of a suitable functional. By studying a corresponding Euler–Lagrange equation, we also determine infinitely many sign-changing solutions at higher energy levels in addition to the found ground-state solutions